Appl Math Optim (2008) 57: 371–400
Boundary Asymptotic Analysis for an Incompressible
Viscous Flow: Navier Wall Laws
M. El Jarroudi · A. Brillard
Published online: 31 October 2007
© Springer Science+Business Media, LLC 2007
Abstract We consider a new way of establishing Navier wall laws. Considering a
bounded domain of R
, N = 2, 3, surrounded by a thin layer
, along a part
of its boundary ∂, we consider a Navier-Stokes ﬂow in ∪∂∪
number of order 1/ε in
.Using-convergence arguments, we describe the asymp-
totic behaviour of the solution of this problem and get a general Navier law involving
a matrix of Borel measures having the same support contained in the interface
We then consider two special cases where we characterize this matrix of measures.
As a further application, we consider an optimal control problem within this context.
Keywords Navier law · Navier-Stokes ﬂow · -convergence · Asymptotic
behaviour · Optimal control problem
A common hypothesis used in ﬂuid mechanics is that, at the interface between a solid
and a ﬂuid, the velocity u of the ﬂuid is equal to that of the solid. If the solid is at
rest, the velocity of the ﬂuid must thus vanish: u = 0, on the boundary of the solid.
These are the so-called rigid boundary conditions. When writing this condition, one
assumes that the ﬂuid perfectly adheres to the solid.
This hypothesis has not always been accepted for a viscous ﬂuid, although some
veriﬁcations have been made through experiments. G. Taylor indeed veriﬁed in 1923
M. El Jarroudi
Département de Mathématiques, FST Tanger, Université Abdelmalek Essaâdi, B.P. 416, Tanger,
A. Brillard (
Laboratoire de Gestion des Risques et Environnement, Université de Haute-Alsace,
25 rue de Chemnitz, 68200 Mulhouse, France